Models dimensionality

Proc., 65(3), 45 (1986)] is preferred. To use this and alternate models, dimensional characteristics of structured packing must be defined. Figure 14-51 shows nomenclature and definitions of key dimensions. Not shown, but also important, is the angle the corrugations make with the horizontal (usu y 45 or 60°). Then the Rocha et al. predictive equation is ... 

In summary, the methods of theoretical and effective capacity estimation of C02 storage comprise volumetric and compressibility methods, flow mathematical and simulation models, dimensional analysis, analytical investigation and Japanese/Chinese methodology. 

Figure 1. Fuel-cell schematic showing the different model dimensionalities. 0-D modeis are simpie equations and are not shown, the 1-D models comprise the sandwich (z direction), the 2-D modeis comprise the 1-D sandwich and either of the two other coordinate directions x or y), and the 3-D models comprise aii three coordinate directions.

The model discretization or the number of collocation points necessary for accurate representation of the profiles within the reactor bed has a major effect on the dimensionality and thus the solution time of the resulting model. As previously discussed, radial collocation with one interior collocation point generally adequately accounts for radial thermal gradients without increasing the dimensionality of the system. However, multipoint radial collocation may be necessary to describe radial concentration profiles. The analysis of Section VI,E shows that, even with very high radial mass Peclet numbers, the radial concentration is nearly uniform and that the axial bulk concentration and radial and axial temperatures are nearly unaffected by assuming uniform radial concentration. Thus model dimensionality can be kept to a minimum by also performing the radial concentration collocation with one interior collocation point. 

In any case, the cross-validation process is repeated a number of times and the squared prediction errors are summed. This leads to a statistic [predicted residual sum of squares (PRESS), the sum of the squared errors] that varies as a function of model dimensionality. Typically a graph (PRESS plot) is used to draw conclusions. The best number of components is the one that minimises the overall prediction error (see Figure 4.16). Sometimes it is possible (depending on the software you can handle) to visualise in detail how the samples behaved in the LOOCV process and, thus, detect if some sample can be considered an outlier (see Figure 4.16a). Although Figure 4.16b is close to an ideal situation because the first minimum is very well defined, two different situations frequently occur ... 

Figure 4.16 Typical example of a PRESS plot to select the model dimensionality (data shown in Figure 4.9, mean centred) (a) overall PRESS and (b) behaviour of each sample for each number of factors considered in the model. Four factors seem to be the optimal choice here as the minimum is clearly defined. 

Fig. 3. (a) Hierarchy of deterministic, continuum models. Dimensional analysis and symmetry are powerful concepts in reducing the dimensionality of complex models, (b) Hierarchy of stochastic models for chemically reacting well-mixed systems. 

It is necessary to decide the appropriate number of components to use in a model. The appropriate dimensionality of a model may even change depending on what the specified purpose of the model is. Hence, appropriate dimensionality of, e.g., a PARAFAC model, is not necessarily identical to the three-way pseudo-rank of the data array. Appropriate model dimensionality is not only a function of the data but also a function of the context and aim of the analysis. Hence, a suitable PARAFAC model for exploring a data set may have a rank different from a PARAFAC model where the scores are used for a subsequent regression model. 

Finally the statement by dementi et al. [190, 223] PLS is claimed to be a more adequate method than multiple regression analysis (MRA) in many respects. In MRA the number of objects should be three times higher than the number of variables, the model dimensionality is fixed a priori, the model is based on the additivity of the effects and the grouping of objects is not detected. 

Data may be presented directly or in summarized form, such as on maps and graphs. However, since humans respond visually in different ways to different geometric forms and arrays, a scientifically correct diagram may sometimes be misleading. Care is therefore required to ensure that the interpretative materials convey exactly what is intended. Large data sets are sometimes reduced to small sets with the aid of empirical or physical models. Dimensional analysis often permits several variables to be collapsed to a single new parameter. In this connection, it is important to note that empirical models caimot be extrapolated with assurance to new situations [50,51]. 

Parameters L.V. Geometry Model Dimensional- ity Type of Deformation Type of Loading (Pressure)... 

For completeness we mention a third method of modeling, dimensional analysis. In Chapter 5 we will explore methods to scale constrained laws or empirical data to dynamically similar systems. For example, the data shown in Figure 4.1 for one particular gas can be used to predict the results for another gas. Whereas a plot such as shown in Figures 3.32 and 4.1 contain data for a specific gas, dimensional analysis allows one to apply these data to any gas, with appropriate scaling of the volume, temperature, and pressure. 

One often speaks in this context about the curse of dimensionality, by making the model slightly more complex, one increases the number of model parameters significantly. One should therefore watch out for models that produce a good model fit due to a high model dimensionality ... 

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